How to convert object space normals to tangent space?
The simplest way to do such a conversion is to calculate the dot products between the Object_Space_Normal and the Tangent plane, and then 'normalize' the result to RGB.
$\vec{Tg}_N = 0.5 +
\begin{bmatrix} \vec{Tg} \cdot \vec{N}_{obj} \\
(\vec{N} \times \vec{Tg}) \cdot N_{obj} \\
\vec{N} \cdot \vec{N}_{obj} \end{bmatrix}
/ 2 $
And with nodes:

This is the main process when working with vectors in World space.
In the case where your Normals are Object normals encoded into RGB, then you need to convert them first to world as in the following example: